Problem: Simplify the following expression: $y = \dfrac{-8x^2- 15x+2}{x + 2}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(2)} &=& -16 \\ {a} + {b} &=& &=& {-15} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-16$ and add them together. Remember, since $-16$ is negative, one of the factors must be negative. The factors that add up to ${-15}$ will be your ${a}$ and ${b}$ When ${a}$ is ${1}$ and ${b}$ is ${-16}$ $ \begin{eqnarray} {ab} &=& ({1})({-16}) &=& -16 \\ {a} + {b} &=& {1} + {-16} &=& -15 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-8}x^2 +{1}x) + ({-16}x +{2}) $ Factor out the common factors: $ x(-8x + 1) + 2(-8x + 1)$ Now factor out $(-8x + 1)$ $ (-8x + 1)(x + 2)$ The original expression can therefore be written: $ \dfrac{(-8x + 1)(x + 2)}{x + 2}$ We are dividing by $x + 2$ , so $x + 2 \neq 0$ Therefore, $x \neq -2$ This leaves us with $-8x + 1; x \neq -2$.